crypto/bn/bn_prime.c

   1 /*
   2  * Copyright 1995-2019 The OpenSSL Project Authors. All Rights Reserved.
   3  *
   4  * Licensed under the Apache License 2.0 (the "License").  You may not use
   5  * this file except in compliance with the License.  You can obtain a copy
   6  * in the file LICENSE in the source distribution or at
   7  * https://www.openssl.org/source/license.html
   8  */
   9
  10 #include <stdio.h>
  11 #include <time.h>
  12 #include "internal/cryptlib.h"
  13 #include "bn_local.h"
  14
  15 /*
  16  * The quick sieve algorithm approach to weeding out primes is Philip
  17  * Zimmermann's, as implemented in PGP.  I have had a read of his comments
  18  * and implemented my own version.
  19  */
  20 #include "bn_prime.h"
  21
  22 static int probable_prime(BIGNUM *rnd, int bits, int safe, prime_t *mods,
  23                           BN_CTX *ctx);
  24 static int probable_prime_dh(BIGNUM *rnd, int bits, int safe, prime_t *mods,
  25                              const BIGNUM *add, const BIGNUM *rem,
  26                              BN_CTX *ctx);
  27
  28 #define square(x) ((BN_ULONG)(x) * (BN_ULONG)(x))
  29
  30 #if BN_BITS2 == 64
  31 # define BN_DEF(lo, hi) (BN_ULONG)hi<<32|lo
  32 #else
  33 # define BN_DEF(lo, hi) lo, hi
  34 #endif
  35
  36 /*
  37  * See SP800 89 5.3.3 (Step f)
  38  * The product of the set of primes ranging from 3 to 751
  39  * Generated using process in test/bn_internal_test.c test_bn_small_factors().
  40  * This includes 751 (which is not currently included in SP 800-89).
  41  */
  42 static const BN_ULONG small_prime_factors[] = {
  43     BN_DEF(0x3ef4e3e1, 0xc4309333), BN_DEF(0xcd2d655f, 0x71161eb6),
  44     BN_DEF(0x0bf94862, 0x95e2238c), BN_DEF(0x24f7912b, 0x3eb233d3),
  45     BN_DEF(0xbf26c483, 0x6b55514b), BN_DEF(0x5a144871, 0x0a84d817),
  46     BN_DEF(0x9b82210a, 0x77d12fee), BN_DEF(0x97f050b3, 0xdb5b93c2),
  47     BN_DEF(0x4d6c026b, 0x4acad6b9), BN_DEF(0x54aec893, 0xeb7751f3),
  48     BN_DEF(0x36bc85c4, 0xdba53368), BN_DEF(0x7f5ec78e, 0xd85a1b28),
  49     BN_DEF(0x6b322244, 0x2eb072d8), BN_DEF(0x5e2b3aea, 0xbba51112),
  50     BN_DEF(0x0e2486bf, 0x36ed1a6c), BN_DEF(0xec0c5727, 0x5f270460),
  51     (BN_ULONG)0x000017b1
  52 };
  53
  54 #define BN_SMALL_PRIME_FACTORS_TOP OSSL_NELEM(small_prime_factors)
  55 static const BIGNUM _bignum_small_prime_factors = {
  56     (BN_ULONG *)small_prime_factors,
  57     BN_SMALL_PRIME_FACTORS_TOP,
  58     BN_SMALL_PRIME_FACTORS_TOP,
  59     0,
  60     BN_FLG_STATIC_DATA
  61 };
  62
  63 const BIGNUM *bn_get0_small_factors(void)
  64 {
  65     return &_bignum_small_prime_factors;
  66 }
  67
  68 /*
  69  * Calculate the number of trial divisions that gives the best speed in
  70  * combination with Miller-Rabin prime test, based on the sized of the prime.
  71  */
  72 static int calc_trial_divisions(int bits)
  73 {
  74     if (bits <= 512)
  75         return 64;
  76     else if (bits <= 1024)
  77         return 128;
  78     else if (bits <= 2048)
  79         return 384;
  80     else if (bits <= 4096)
  81         return 1024;
  82     return NUMPRIMES;
  83 }
  84
  85 int BN_GENCB_call(BN_GENCB *cb, int a, int b)
  86 {
  87     /* No callback means continue */
  88     if (!cb)
  89         return 1;
  90     switch (cb->ver) {
  91     case 1:
  92         /* Deprecated-style callbacks */
  93         if (!cb->cb.cb_1)
  94             return 1;
  95         cb->cb.cb_1(a, b, cb->arg);
  96         return 1;
  97     case 2:
  98         /* New-style callbacks */
  99         return cb->cb.cb_2(a, b, cb);
 100     default:
 101         break;
 102     }
 103     /* Unrecognised callback type */
 104     return 0;
 105 }
 106
 107 int BN_generate_prime_ex2(BIGNUM *ret, int bits, int safe,
 108                           const BIGNUM *add, const BIGNUM *rem, BN_GENCB *cb,
 109                           BN_CTX *ctx)
 110 {
 111     BIGNUM *t;
 112     int found = 0;
 113     int i, j, c1 = 0;
 114     prime_t *mods = NULL;
 115     int checks = BN_prime_checks_for_size(bits);
 116
 117     if (bits < 2) {
 118         /* There are no prime numbers this small. */
 119         BNerr(BN_F_BN_GENERATE_PRIME_EX2, BN_R_BITS_TOO_SMALL);
 120         return 0;
 121     } else if (add == NULL && safe && bits < 6 && bits != 3) {
 122         /*
 123          * The smallest safe prime (7) is three bits.
 124          * But the following two safe primes with less than 6 bits (11, 23)
 125          * are unreachable for BN_rand with BN_RAND_TOP_TWO.
 126          */
 127         BNerr(BN_F_BN_GENERATE_PRIME_EX2, BN_R_BITS_TOO_SMALL);
 128         return 0;
 129     }
 130
 131     mods = OPENSSL_zalloc(sizeof(*mods) * NUMPRIMES);
 132     if (mods == NULL)
 133         goto err;
 134
 135     BN_CTX_start(ctx);
 136     t = BN_CTX_get(ctx);
 137     if (t == NULL)
 138         goto err;
 139  loop:
 140     /* make a random number and set the top and bottom bits */
 141     if (add == NULL) {
 142         if (!probable_prime(ret, bits, safe, mods, ctx))
 143             goto err;
 144     } else {
 145         if (!probable_prime_dh(ret, bits, safe, mods, add, rem, ctx))
 146             goto err;
 147     }
 148
 149     if (!BN_GENCB_call(cb, 0, c1++))
 150         /* aborted */
 151         goto err;
 152
 153     if (!safe) {
 154         i = BN_is_prime_fasttest_ex(ret, checks, ctx, 0, cb);
 155         if (i == -1)
 156             goto err;
 157         if (i == 0)
 158             goto loop;
 159     } else {
 160         /*
 161          * for "safe prime" generation, check that (p-1)/2 is prime. Since a
 162          * prime is odd, We just need to divide by 2
 163          */
 164         if (!BN_rshift1(t, ret))
 165             goto err;
 166
 167         for (i = 0; i < checks; i++) {
 168             j = BN_is_prime_fasttest_ex(ret, 1, ctx, 0, cb);
 169             if (j == -1)
 170                 goto err;
 171             if (j == 0)
 172                 goto loop;
 173
 174             j = BN_is_prime_fasttest_ex(t, 1, ctx, 0, cb);
 175             if (j == -1)
 176                 goto err;
 177             if (j == 0)
 178                 goto loop;
 179
 180             if (!BN_GENCB_call(cb, 2, c1 - 1))
 181                 goto err;
 182             /* We have a safe prime test pass */
 183         }
 184     }
 185     /* we have a prime :-) */
 186     found = 1;
 187  err:
 188     OPENSSL_free(mods);
 189     BN_CTX_end(ctx);
 190     bn_check_top(ret);
 191     return found;
 192 }
 193
 194 #ifndef FIPS_MODE
 195 int BN_generate_prime_ex(BIGNUM *ret, int bits, int safe,
 196                          const BIGNUM *add, const BIGNUM *rem, BN_GENCB *cb)
 197 {
 198     BN_CTX *ctx = BN_CTX_new();
 199     int retval;
 200
 201     if (ctx == NULL)
 202         return 0;
 203
 204     retval = BN_generate_prime_ex2(ret, bits, safe, add, rem, cb, ctx);
 205
 206     BN_CTX_free(ctx);
 207     return retval;
 208 }
 209 #endif
 210
 211 int BN_is_prime_ex(const BIGNUM *a, int checks, BN_CTX *ctx_passed,
 212                    BN_GENCB *cb)
 213 {
 214     return BN_is_prime_fasttest_ex(a, checks, ctx_passed, 0, cb);
 215 }
 216
 217 /* See FIPS 186-4 C.3.1 Miller Rabin Probabilistic Primality Test. */
 218 int BN_is_prime_fasttest_ex(const BIGNUM *w, int checks, BN_CTX *ctx,
 219                             int do_trial_division, BN_GENCB *cb)
 220 {
 221     int i, status, ret = -1;
 222 #ifndef FIPS_MODE
 223     BN_CTX *ctxlocal = NULL;
 224 #else
 225
 226     if (ctx == NULL)
 227         return -1;
 228 #endif
 229
 230     /* w must be bigger than 1 */
 231     if (BN_cmp(w, BN_value_one()) <= 0)
 232         return 0;
 233
 234     /* w must be odd */
 235     if (BN_is_odd(w)) {
 236         /* Take care of the really small prime 3 */
 237         if (BN_is_word(w, 3))
 238             return 1;
 239     } else {
 240         /* 2 is the only even prime */
 241         return BN_is_word(w, 2);
 242     }
 243
 244     /* first look for small factors */
 245     if (do_trial_division) {
 246         int trial_divisions = calc_trial_divisions(BN_num_bits(w));
 247
 248         for (i = 1; i < trial_divisions; i++) {
 249             BN_ULONG mod = BN_mod_word(w, primes[i]);
 250             if (mod == (BN_ULONG)-1)
 251                 return -1;
 252             if (mod == 0)
 253                 return BN_is_word(w, primes[i]);
 254         }
 255         if (!BN_GENCB_call(cb, 1, -1))
 256             return -1;
 257     }
 258 #ifndef FIPS_MODE
 259     if (ctx == NULL && (ctxlocal = ctx = BN_CTX_new()) == NULL)
 260         goto err;
 261 #endif
 262
 263     ret = bn_miller_rabin_is_prime(w, checks, ctx, cb, 0, &status);
 264     if (!ret)
 265         goto err;
 266     ret = (status == BN_PRIMETEST_PROBABLY_PRIME);
 267 err:
 268 #ifndef FIPS_MODE
 269     BN_CTX_free(ctxlocal);
 270 #endif
 271     return ret;
 272 }
 273
 274 /*
 275  * Refer to FIPS 186-4 C.3.2 Enhanced Miller-Rabin Probabilistic Primality Test.
 276  * OR C.3.1 Miller-Rabin Probabilistic Primality Test (if enhanced is zero).
 277  * The Step numbers listed in the code refer to the enhanced case.
 278  *
 279  * if enhanced is set, then status returns one of the following:
 280  *     BN_PRIMETEST_PROBABLY_PRIME
 281  *     BN_PRIMETEST_COMPOSITE_WITH_FACTOR
 282  *     BN_PRIMETEST_COMPOSITE_NOT_POWER_OF_PRIME
 283  * if enhanced is zero, then status returns either
 284  *     BN_PRIMETEST_PROBABLY_PRIME or
 285  *     BN_PRIMETEST_COMPOSITE
 286  *
 287  * returns 0 if there was an error, otherwise it returns 1.
 288  */
 289 int bn_miller_rabin_is_prime(const BIGNUM *w, int iterations, BN_CTX *ctx,
 290                              BN_GENCB *cb, int enhanced, int *status)
 291 {
 292     int i, j, a, ret = 0;
 293     BIGNUM *g, *w1, *w3, *x, *m, *z, *b;
 294     BN_MONT_CTX *mont = NULL;
 295
 296     /* w must be odd */
 297     if (!BN_is_odd(w))
 298         return 0;
 299
 300     BN_CTX_start(ctx);
 301     g = BN_CTX_get(ctx);
 302     w1 = BN_CTX_get(ctx);
 303     w3 = BN_CTX_get(ctx);
 304     x = BN_CTX_get(ctx);
 305     m = BN_CTX_get(ctx);
 306     z = BN_CTX_get(ctx);
 307     b = BN_CTX_get(ctx);
 308
 309     if (!(b != NULL
 310             /* w1 := w - 1 */
 311             && BN_copy(w1, w)
 312             && BN_sub_word(w1, 1)
 313             /* w3 := w - 3 */
 314             && BN_copy(w3, w)
 315             && BN_sub_word(w3, 3)))
 316         goto err;
 317
 318     /* check w is larger than 3, otherwise the random b will be too small */
 319     if (BN_is_zero(w3) || BN_is_negative(w3))
 320         goto err;
 321
 322     /* (Step 1) Calculate largest integer 'a' such that 2^a divides w-1 */
 323     a = 1;
 324     while (!BN_is_bit_set(w1, a))
 325         a++;
 326     /* (Step 2) m = (w-1) / 2^a */
 327     if (!BN_rshift(m, w1, a))
 328         goto err;
 329
 330     /* Montgomery setup for computations mod a */
 331     mont = BN_MONT_CTX_new();
 332     if (mont == NULL || !BN_MONT_CTX_set(mont, w, ctx))
 333         goto err;
 334
 335     if (iterations == BN_prime_checks)
 336         iterations = BN_prime_checks_for_size(BN_num_bits(w));
 337
 338     /* (Step 4) */
 339     for (i = 0; i < iterations; ++i) {
 340         /* (Step 4.1) obtain a Random string of bits b where 1 < b < w-1 */
 341         if (!BN_priv_rand_range_ex(b, w3, ctx)
 342                 || !BN_add_word(b, 2)) /* 1 < b < w-1 */
 343             goto err;
 344
 345         if (enhanced) {
 346             /* (Step 4.3) */
 347             if (!BN_gcd(g, b, w, ctx))
 348                 goto err;
 349             /* (Step 4.4) */
 350             if (!BN_is_one(g)) {
 351                 *status = BN_PRIMETEST_COMPOSITE_WITH_FACTOR;
 352                 ret = 1;
 353                 goto err;
 354             }
 355         }
 356         /* (Step 4.5) z = b^m mod w */
 357         if (!BN_mod_exp_mont(z, b, m, w, ctx, mont))
 358             goto err;
 359         /* (Step 4.6) if (z = 1 or z = w-1) */
 360         if (BN_is_one(z) || BN_cmp(z, w1) == 0)
 361             goto outer_loop;
 362         /* (Step 4.7) for j = 1 to a-1 */
 363         for (j = 1; j < a ; ++j) {
 364             /* (Step 4.7.1 - 4.7.2) x = z. z = x^2 mod w */
 365             if (!BN_copy(x, z) || !BN_mod_mul(z, x, x, w, ctx))
 366                 goto err;
 367             /* (Step 4.7.3) */
 368             if (BN_cmp(z, w1) == 0)
 369                 goto outer_loop;
 370             /* (Step 4.7.4) */
 371             if (BN_is_one(z))
 372                 goto composite;
 373         }
 374         /* At this point z = b^((w-1)/2) mod w */
 375         /* (Steps 4.8 - 4.9) x = z, z = x^2 mod w */
 376         if (!BN_copy(x, z) || !BN_mod_mul(z, x, x, w, ctx))
 377             goto err;
 378         /* (Step 4.10) */
 379         if (BN_is_one(z))
 380             goto composite;
 381         /* (Step 4.11) x = b^(w-1) mod w */
 382         if (!BN_copy(x, z))
 383             goto err;
 384 composite:
 385         if (enhanced) {
 386             /* (Step 4.1.2) g = GCD(x-1, w) */
 387             if (!BN_sub_word(x, 1) || !BN_gcd(g, x, w, ctx))
 388                 goto err;
 389             /* (Steps 4.1.3 - 4.1.4) */
 390             if (BN_is_one(g))
 391                 *status = BN_PRIMETEST_COMPOSITE_NOT_POWER_OF_PRIME;
 392             else
 393                 *status = BN_PRIMETEST_COMPOSITE_WITH_FACTOR;
 394         } else {
 395             *status = BN_PRIMETEST_COMPOSITE;
 396         }
 397         ret = 1;
 398         goto err;
 399 outer_loop: ;
 400         /* (Step 4.1.5) */
 401         if (!BN_GENCB_call(cb, 1, i))
 402             goto err;
 403     }
 404     /* (Step 5) */
 405     *status = BN_PRIMETEST_PROBABLY_PRIME;
 406     ret = 1;
 407 err:
 408     BN_clear(g);
 409     BN_clear(w1);
 410     BN_clear(w3);
 411     BN_clear(x);
 412     BN_clear(m);
 413     BN_clear(z);
 414     BN_clear(b);
 415     BN_CTX_end(ctx);
 416     BN_MONT_CTX_free(mont);
 417     return ret;
 418 }
 419
 420 /*
 421  * Generate a random number of |bits| bits that is probably prime by sieving.
 422  * If |safe| != 0, it generates a safe prime.
 423  * |mods| is a preallocated array that gets reused when called again.
 424  *
 425  * The probably prime is saved in |rnd|.
 426  *
 427  * Returns 1 on success and 0 on error.
 428  */
 429 static int probable_prime(BIGNUM *rnd, int bits, int safe, prime_t *mods,
 430                           BN_CTX *ctx)
 431 {
 432     int i;
 433     BN_ULONG delta;
 434     int trial_divisions = calc_trial_divisions(bits);
 435     BN_ULONG maxdelta = BN_MASK2 - primes[trial_divisions - 1];
 436
 437  again:
 438     /* TODO: Not all primes are private */
 439     if (!BN_priv_rand_ex(rnd, bits, BN_RAND_TOP_TWO, BN_RAND_BOTTOM_ODD, ctx))
 440         return 0;
 441     if (safe && !BN_set_bit(rnd, 1))
 442         return 0;
 443     /* we now have a random number 'rnd' to test. */
 444     for (i = 1; i < trial_divisions; i++) {
 445         BN_ULONG mod = BN_mod_word(rnd, (BN_ULONG)primes[i]);
 446         if (mod == (BN_ULONG)-1)
 447             return 0;
 448         mods[i] = (prime_t) mod;
 449     }
 450     delta = 0;
 451  loop:
 452     for (i = 1; i < trial_divisions; i++) {
 453         /*
 454          * check that rnd is a prime and also that
 455          * gcd(rnd-1,primes) == 1 (except for 2)
 456          * do the second check only if we are interested in safe primes
 457          * in the case that the candidate prime is a single word then
 458          * we check only the primes up to sqrt(rnd)
 459          */
 460         if (bits <= 31 && delta <= 0x7fffffff
 461                 && square(primes[i]) > BN_get_word(rnd) + delta)
 462             break;
 463         if (safe ? (mods[i] + delta) % primes[i] <= 1
 464                  : (mods[i] + delta) % primes[i] == 0) {
 465             delta += safe ? 4 : 2;
 466             if (delta > maxdelta)
 467                 goto again;
 468             goto loop;
 469         }
 470     }
 471     if (!BN_add_word(rnd, delta))
 472         return 0;
 473     if (BN_num_bits(rnd) != bits)
 474         goto again;
 475     bn_check_top(rnd);
 476     return 1;
 477 }
 478
 479 /*
 480  * Generate a random number |rnd| of |bits| bits that is probably prime
 481  * and satisfies |rnd| % |add| == |rem| by sieving.
 482  * If |safe| != 0, it generates a safe prime.
 483  * |mods| is a preallocated array that gets reused when called again.
 484  *
 485  * Returns 1 on success and 0 on error.
 486  */
 487 static int probable_prime_dh(BIGNUM *rnd, int bits, int safe, prime_t *mods,
 488                              const BIGNUM *add, const BIGNUM *rem,
 489                              BN_CTX *ctx)
 490 {
 491     int i, ret = 0;
 492     BIGNUM *t1;
 493     BN_ULONG delta;
 494     int trial_divisions = calc_trial_divisions(bits);
 495     BN_ULONG maxdelta = BN_MASK2 - primes[trial_divisions - 1];
 496
 497     BN_CTX_start(ctx);
 498     if ((t1 = BN_CTX_get(ctx)) == NULL)
 499         goto err;
 500
 501     if (maxdelta > BN_MASK2 - BN_get_word(add))
 502         maxdelta = BN_MASK2 - BN_get_word(add);
 503
 504  again:
 505     if (!BN_rand_ex(rnd, bits, BN_RAND_TOP_ONE, BN_RAND_BOTTOM_ODD, ctx))
 506         goto err;
 507
 508     /* we need ((rnd-rem) % add) == 0 */
 509
 510     if (!BN_mod(t1, rnd, add, ctx))
 511         goto err;
 512     if (!BN_sub(rnd, rnd, t1))
 513         goto err;
 514     if (rem == NULL) {
 515         if (!BN_add_word(rnd, safe ? 3u : 1u))
 516             goto err;
 517     } else {
 518         if (!BN_add(rnd, rnd, rem))
 519             goto err;
 520     }
 521
 522     if (BN_num_bits(rnd) < bits
 523             || BN_get_word(rnd) < (safe ? 5u : 3u)) {
 524         if (!BN_add(rnd, rnd, add))
 525             goto err;
 526     }
 527
 528     /* we now have a random number 'rnd' to test. */
 529     for (i = 1; i < trial_divisions; i++) {
 530         BN_ULONG mod = BN_mod_word(rnd, (BN_ULONG)primes[i]);
 531         if (mod == (BN_ULONG)-1)
 532             goto err;
 533         mods[i] = (prime_t) mod;
 534     }
 535     delta = 0;
 536  loop:
 537     for (i = 1; i < trial_divisions; i++) {
 538         /* check that rnd is a prime */
 539         if (bits <= 31 && delta <= 0x7fffffff
 540                 && square(primes[i]) > BN_get_word(rnd) + delta)
 541             break;
 542         /* rnd mod p == 1 implies q = (rnd-1)/2 is divisible by p */
 543         if (safe ? (mods[i] + delta) % primes[i] <= 1
 544                  : (mods[i] + delta) % primes[i] == 0) {
 545             delta += BN_get_word(add);
 546             if (delta > maxdelta)
 547                 goto again;
 548             goto loop;
 549         }
 550     }
 551     if (!BN_add_word(rnd, delta))
 552         goto err;
 553     ret = 1;
 554
 555  err:
 556     BN_CTX_end(ctx);
 557     bn_check_top(rnd);
 558     return ret;
 559 }